Shortcomings of UNIFAC-FV to characterize the phase behavior of

284

I n d . Eng. Chem. Res. 1988,27, 284-294

MATERIALS AND INTERFACES Shortcomings of UNIFAC-FV To Characterize the Phase Behavior of Polymer-Polymer Blends Laurence A. Belfiore,* Ashutosh A. Patwardhan,+and Terry G. Lenz Department of Agricultural and Chemical Engineering, Colorado State University, Fort Collins, Colorado 80523

The concentration dependence of the Gibbs free energy of mixing was predicted for binary mixtures of chain molecules. T o accomplish this task, the UNIFAC-FV group contribution model of Oishi and Prausnitz for polymer solutions was extended to polymer-polymer blends. Modifications of the combinatorial and free-volume contributions to the activity of polymeric component i, implemented herein, account for the chainlike nature of both components in the mixtures. Thermodynamic group-contribution predictions for blends of polystyrene and poly(viny1 methyl ether) are consistent with experimental miscibility results (via cloud-point measurements) below the lower critical solution temperature phenomenon. However, UNIFAC-FV fails to predict the well-known concentration and molecular weight dependent phase instability of these blends at elevated temperatures. Model predictions of the phase stability/instability for blends of polystyrene with poly(methy1 acrylate) and poly(ethy1ene glycol) with poly(acry1ic acid) contradict experimental results in the literature, as well as the calorimetric data presented herein. To explain these shortcomings, the residual contribution to the activity of polymeric component i, using functional group interaction parameters appropriate to small molecules, was found to be exaggerated in magnitude for some polymer-polymer blends. This flaw in the UNIFAC-FV scheme precludes an accurate prediction of the phase behavior of chain-molecule mixtures in their equilibrium amorphous state. Discrepancies between prediction and experiment for the above-mentioned blends should be interpreted with caution, however, due to the “equilibrium” nature of the UNIFAC-FV predictions and the “kinetic” nature of polymerpolymer blends.

Introduction Previous publications by Oishi and Prausnitz (1978) and Patwardhan and Belfiore (1986) illustrate that the UNIFAC-FV model (Universal Quasi-Chemical FunctionalGroup Activity Coefficient formalism which includes a Free-Volume correction for species of different molecular size) predicts the thermodynamic properties of polymer solutions quite well. A logical extension of this group contribution approach is an application of the model to predict the equilibrium phase behavior of mixtures of dissimilar chain molecules (i.e., moderate molecular weight polymer-polymer blends) via the concentration dependence of the Gibbs free energy of mixing. However, one should realize that the UNIFAC group parameters were determined from equilibrium thermodynamic measurements on “small”-moleculeliquid mixtures (Abrams and Prausnitz, 1975; Fredenslund et al., 1977). In this respect, an application of UNIFAC-FV to polymer-polymer blends should provide a severe test of this predictive scheme in a molecular motion regime where the functional group parameters are, most likely, not expected to model the phase behavior observed experimentally. Furthermore, one might question the “equilibrium” nature of the observed phase behavior for mixtures of chain molecules due to the sluggishness of the material’s viscoelastic response. This Present address: Department of Chemical Engineering, University of Texas, Austin, TX 78712. 0888-5885/88/2627-0284$01.50/0

concern should preclude an exact interpretation of binary polymer blend phase diagrams in terms of equilibrium principles (Belfiore and Cooper, 1983). The thermodynamic group contribution calculations presented herein for blends of atactic polystyrene (PSI and poly(viny1 methyl ether) (PVME) are qualitatively consistent with classical experimental results (Kwei et al., 1974; Nishi et al., 1975) demonstrating complete miscibility for this mixture (based on the concentration dependence of the Gibbs free energy of mixing) at temperatures below the lower critical solution phenomenon, Furthermore, our predictions of the Flory-Huggins thermodyqamic interaction parameter x ’ s p m (per repeat unit of polystyrene), although slightly larger in absolute magnitude than the most recent calculations of x’ by Hadziioannou and Stein (1984) (via small-angle neutron scattering), are negative and well within the range of previous x’ values for PSPVME blends determined from experimental thermodynamic studies. On the other hand, UNIFAC-FV calculations for chain-molecule mixtures of atactic polystyrene and poly(methy1acrylate) do not agree with previous experimental data (via melt titrations) in the literature (Somani and Shaw, 1981). Calorimetric results presented herein (via DSC) have identified discrepancies between UNIFAC-FV and experiments for a second kind of blends [e.g., poly(ethy1ene glycol) and poly(acry1ic acid)] in which the solid-state phase behavior is governed by the propensity for intermolecular hydrogen bonding (Smith et al., 1959) versus the driving force for poly(ethy1ene glycol) 0 1988 American Chemical Society

Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 285 crystallization (Belfiore, 1987). We track the problem in predicting these well-documented experimental results to the residual contribution to the polymer activity in the UNIFAC scheme and suggest reasons for the failure of this group contribution correlation to model accurately the energetic interactions between dissimilar chain molecules.

UNIFAC-FV Model The UNIFAC-FV model partitions the activity of component 1 in a binary mixture as follows: + In alFV + In alR (1) In al = In alc combinatorial freeresidual volume The contribution to the activity due to differences in shapes and sizes of the two dissimilar molecules is contained in the combinatorial part. This is primarily an entropic contribution which incorporates Staverman's (1950) combinatorial formula for mixtures of molecules of arbitrary size and shape. Attractive forces between the two kinds of molecules are not included in this entropic term. The expression for the combinatorial activity of component 1 is In alc =

41

In 41 + 1 - - + '/ZMlql{ln X1

(W41)- (1 - @l/f4)1(2)

Here, Ml is the molecular weight of component 1 , Z is the lattice coordination number (set equal to lo),x1 is the mole fraction of component 1 in the mixture, and O1 and 41are the surface fraction and segment fraction of component 1, respectively. For binary mixtures, O1 and $1 are defined by 41Wl rlwl 81 = 41 = (3) %Wl+ q2% r1w1 + rzwz where wi is the weight fraction of species i in the mixture. The pure-component parameters ri and qi in eq 3 represent the van der Waals volume and surface area per unit mass, respectively. They are calculated in the following manner:

where vk(i) is the number of functional groups of type k in a molecule of species i. The molar group volume Rk and area parameter Qk have been tabulated by Gmehling et al. (1982) and may be obtained from the van der Waals group volume vwkand surface area Awk given by Bondi (1968) as Rk = Vwk/15.17

Qk = A,k/(2.5 x 10')

(5) lo9,are calcu-

The normalization factors, 15.17 and 2.5 X lated by Abrams and Prausnitz (1975). The free volume term accounts for changes in unoccupied volume caused by mixing. The effect of including this contribution in the expression for the activity of one of the components (see eq 1) ensures one that the Flory-Huggins intermolecular interaction parameter, x, or, rather, the dimensionless interaction free energy of mixing will exhibit concentration dependence (Bates, 1985). Group contribution predictions of the concentration dependence of the x parameter have been demonstrated for the following classical solutions: poly(isobuty1ene) in either benzene or n-pentane and poly(styrene) in toluene (Patwardhan and Belfiore, 1986). The expression for the free-volume term is obtained from a simplification of Flory's equation-ofstate approach (Flory, 1970). If one sets the difference between the potential energies of dissimilar molecules

(components 1 and 2) to zero (this realistic non-zero difference is accounted for by the residual part of UNIFAC), then the free-volume contribution becomes (Oishi and Prausnitz, 1978) In alFV='

with

Here, Viis the volume per gram of component i. It is calculated using the specific volume (or density) of the component at an arbitrary reference temperature (usually 4 or 20 O C ) together with its coefficient of thermal expansion. These data are readily available for many polymers and solvents (Brandrup and Immergut, 1975). The constant C1 is related to the number of external degrees of freedom per molecule of component 1 (Bonner and Prausnitz, 1973). For common solvents with molecular weights in the vicinity of 100, C1has been assigned a value of 1.1 (Oishi and Prausnitz, 1978). For solvents of significantly higher molecular weight, a larger value of C1 should be used. A comparison between calculated and experimental activities for solvents in polymer solutions has indicated that agreement is best when b = 1.28 in eq 7 (Oishi and Prausnitz, 1978). The residual part of the activity from eq 1 is due to both intermolecular and intramolecular interactions and includes the effects of both dispersion forces and, to some extent, hydrogen-bonding-like interactions between molecules. The functional form of the residual activity is derived using the solution-of-groups concept and is written as (Oishi and Prausnitz, 1978) In alR =

C

vk'"(h

k all groups

rk - In rk"))

(8)

where rk is the residual activity of group k in the mixture and I',(l) is the residual activity of group k in a reference solution containing molecules of component 1 only. Following Oishi and Prausnitz (1978), the residual activity of group k in the mixture is expressed as In r k = & k [ l - In (CHm$"k) - C(Hm$km/(CHn$'nm))I m

m

n

(9)

where the sums are over all groups, and Mk is the molecular weight of group k. The area fraction Hmof group m is calculated by using the equation QmXm

H, = XQnXn n

where x , is the mole fraction of group m in the mixture. The quantity Gmn in eq 9 contains interaction energies between the various functional groups (both intramolecular and intermolecular) in the mixture. It is calculated from a Boltzmann-type equation: 9 m n = exP(-amn/ T ) (11) where the group interaction parameter amnis evaluated from a data base using a wide range of experimental results (Fredenslund et al., 1977; Gmehling et al., 1982; Macedo et d.,1983). r k ( l ) is found in a similar fashion. In this case, however, the sums in eq 9 include only those groups present in a molecule of component 1.

286 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988

Extension of the UNIFAC-FV Model The application of the UNIFAC-FV model to polymer-polymer blends differs from that for polymer solutions in two respects. The first concerns the combinatorial part of the activity. Thomas and Eckert (1984) pointed out that Staverman's (1950) combinatorial formula sometimes leads to combinatorial activities greater than unity, which is a physically unrealizable result. Since the combinatorial entropy of mixing is necessarily positive (Patterson, 1982; Kammer, 1986), the term -TASmiXin the thermodynamic expression for the isothermal Gibbs free energy of mixing (at temperature T), (12) AG,, = AHmix- TAS,ix should always provide a negative contribution to AG,, and favor the mixing process. It should be mentioned that a negative free-energy change upon mixing is necessary but not sufficient for the dissolution process to occur spontaneously. The criterion of chemical or diffusional stability in a binary mixture necessarily requires that

'

((e2/JN12) AGmixJT,p,N2 0 where N 1 and N2 represent mole numbers for components 1 and 2, respectively. Furthermore, Euler's integral theorem for homogeneous thermodynamic functions (to degree n = 1with respect to the molar mass of the system) allows one to express the extensive Gibbs free energy of mixing for a binary system in the following form with the aid of eq 1: AGmi, = N I R q l n alc + In alFV+ In alR) + N2RT(ln aZc + In aZFV+ In azR) (13) For the specific case of a hypothetical athermal mixture of two small molecules (similar in molecular size), the free-volume and residual contributions to the activity are negligible (Weidlich and Gmehling, 1987), and eq 13 reduces to

AGmix= RT(N1In alc

+ N 2 In azC1

(14)

which indicates the dominance of the entropic contribution toward spontaneous mixing. In eq 14, combinatorial activities greater than unity will lead necessarily to a positive free-energy change upon mixing, a result which contradicts completely the notion of entropic stabilization. When eq 2 is used to calculate the combinatorial activity of a polymer in a blend, one frequently finds that either alC,aZc,or both are greater than unity. To overcome this difficulty, Kikic et al. (1980) suggested modifying the definition of the segment fraction in eq 3 in the following manner w,rJ2I3 4L = ___ (15) Cw1r12J3 I

for use in eq 2. Thomas and Eckert (1984) recommended raising the exponent in eq 15 from 0.67 to 0.75 and subsequently dropping those terms in eq 2 preceded by ZMiqi/2 (e.g., the last two terms on the right-hand side of eq 2). Our computational scheme employs the modification proposed by Thomas and Eckert in cases where the Staverman form (eq 2) yields combinatorial activities greater than unity. However, the Staverman form for the combinatorial contribution to the activity is retained when this unrealistic problem is not encountered. In the examples illustrated below, the modification proposed by Thomas and Eckert (1984) was employed for blends of polystyrene with poly(viny1methyl ether) and polystyrene with poly(methy1acrylate). The Staverman expression (eq

Table I. Characteristic Parameters in Flory's Equation-of-State Model Which Allows One To Calculate the Free-Volume Constant (C)for Some Common Polymers (Taken from Bonner and Prausnitz [1973]):

c = -P*V*M RT*

polymer poly(acry1ic acid) poly(ethy1 acrylate) poly(ethy1ene oxide) poly(methy1 methacrylate) poly(n-butyl methacrylate) polystyrene

V*, cm3/g 0.694 0.728 0.753 0.762 0.832 0.817

P*, bar T * , K 102C/Ma 9070 5700 6720 11400 6500

7270 6580 6450 11400 8010

1.04 0.76 0.94 0.92 0.81

5340

7970

0.66

M is the polymer molecular weight, considered to be monodisperse in the calculations discussed in the text.

2) was employed for relatively low molecular weight blends of poly(ethy1ene glycol) (MW = 3400) and poly(acry1ic acid) (MW = 5000). The second change introduced in the UNIFAC-FV model for polymer-polymer blends involves the free-volume constant, C,, which was assigned a value of 1.1 for small molecules (see eq 6, written for component 1 in a binary mixture). The quantity 3Ci is equal to the number of external degrees of freedom per molecule of component i (Bonner and Prausnitz, 1973). It is defined in terms of the characteristic pressure, volume, and temperature (P*, V * ,and T * )of pure component i taken from Flory's (1970) equation-of-state model. Here, V* is the hard-core specific volume of the species of interest, T * is a measure of the potential energy per external degree of freedom, and P * is a characteristic potential energy density. These parameters are tabulated by Bonner and Prausnitz (1973), Beret and Prausnitz (1975), and Somani and Shaw (1981) for a variety of polymers. Values of P * , V*, T*, and the associated value of the free volume constant C, for a few polymers are given in Table I. The appropriate value of C, consistent with the hypothetical monodisperse molecular weight of polymeric component i, in the examples discussed below, is used in eq 6 to calculate the free-volume activity of component i (via UNIFAC-FV) in a polymerpolymer blend.

Results and Discussion Polystyrene-Poly(viny1 methyl ether). The concentration dependence of the specific Gibbs free energy of mixing, predicted by the UNIFAC-FV model for blends of polystyrene and poly(viny1 methyl ether), is shown in Figure 1at temperatures of 100,130,150, and 200 "C. The hypothetical monodisperse molecular weights of PS and PVME were selected to be 51 000 and 51 500, respectively. In this manner, the cloud-point curves for monodisperse PS and polydisperse PVME, generated by Nishi and Kwei (1975), represent experimental results against which the UNIFAC-FB predictions can be tested, particularly in the vicinity of the lower critical solution temperatures. The free-volume contribution to the activity of polystyrene was calculated by setting the free-volume constant C, to the value of 337. Based on the value of Cl/Ml = 6.6 X for polystyrene (see Table I), the free-volume constant Cl (=337), employed in eq 6, was obtained for a hypothetical monodisperse molecular weight of 51 000. The model predictions in Figure 1 were generated as follows: a. With the aid of eq 1-11 (including the modification of the combinatorial activity proposed by Thomas and Eckert (1984))and tabulated functional group parameters for liquid-liquid equilibrium (Magnussen et al., 1981),the

Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 287

I

n

b

0

W

w

II: [L

UI

m

mL1

0.0

0.2

0.4

0.6

0.8

1.a

WEIGHT FR4CTION OF POLYSMiENE

Figure 1. Concentration dependence of the Gibbs free energy of mixing for blends of polystyrene and poly(viny1 methyl ether). Calculations were performed on monodisperse molecular weight polymers PS (51 OOO) and PVME (51 500) at the following temperatures: (a) 100, (b) 130, (c) 150, (d) 200 O C .

activity of polystyrene (PS) (via eq 1)was calculated over the entire concentration range of the binary mixture. b. The activity of poly(viny1methyl ether) (PVME) was determined by using an integral form of the Gibbs-Duhem equation:

Integration was performed numerically as described in a previous publication (Patwardhan and Belfiore, 1986). c. The specific Gibbs free energy of mixing (per unit mass of solution) was calculated via Euler's integral theorem for homogeneous (tothe first degree) thermodynamic functions (Model1 and Reid, 1983):

where w irepresents weight fraction and MWi is the molecular weight of polymeric component i in the mixture. For temperatures between 100 and 200 "C, UNIFAC-FC predictions of the specific Gibbs free energy of mixing in Figure 1are negative and concave upward over the entire concentration range (at constant temperature and pressure). Based on the criteria for diffusional stability in a binary mixture, the curves in Figure 1 are characteristic of a completely miscible blend. This is somewhat consistent with literature citings of the miscibility of these two polymers in the solid state at ambient temperature (Hadziioannou and Stein, 1984; Bank et al., 1971) and at elevated temperatures below the concentration-dependent and molecular weight dependent lower critical solution phenomena (Nishi and Kwei, 1975; Herkt-Maetzky and Schelten, 1983; Kwei et al., 1974; Su and Patterson, 1977). For the particular polymer molecular weights chosen to calculate the Gibbs free energy of mixing via UNIFAC-FV

in blends of polystyrene and poly(viny1methyl ether) (i.e., 51 000 for PS; 51 500 for PVME), Nishi and Kwei (1975) have reported concentration-dependent cloud points ranging between 120 and 150 "C. In a parallel study of the phase behavior of polystyrene (MW = 200 OOO, MJM,, = 1.06) and poly(viny1methyl ether) (aw = 51 500), Nishi et al. (1975) provide evidence that phase separation is initiated (thermally induced) at least 20 "C below the reported lower critical solution temperatures (termed the apparent completion temperature based on transmitted light intensity from thermochromatic studies). In this respect, the UNIFAC-FV predictions of the concentration-dependent Gibbs free energy of mixing (Figure 1) should reveal a concentration region of the incipient instability, particularly for the predictions at 150 and 200 OC. Hence, although the slightly modified UNIFAC-FV formalism described herein predicts the correct equilibrium thermodynamic phase behavior in blends of polystyrene and poly(viny1 methyl ether) at temperatures below the lower critical solution temperature phenomena, it fails to predict the high-temperature phase instability observed experimentally. In summary, the temperature dependence of the equilibrium phase behavior, predicted by UNIFAC-FV, is extremely weak (Leung and Badakhshan, 1987) both for polymer-solvent mixtures (Patwardhan and Belfiore, 1986) and polymer-polymer blends. To the best of our knowledge, upper- (UCST) and lower-critical solution temperatures (LCST) have not been predicted successfully for chain-molecule mixtures via thermodynamic group contribution methods. Temperature effects are embedded in the UNIFAC-FV model described herein (see eq 1-11) via the residual and free-volume contributions to the activity of polymeric component i. Energetic effects (see eq 8-11) allow for temperature dependence via the Boltzmann-like 1c, parameters in eq 11. However, the group interaction parameters am,, in eq 11 are considered, as a first approximation, to be temperature-independent. Skjold-Jerrgensen et al. (1980) have incorporated additional temperature dependence in the residual and combinatorial contributions via the group interaction parameters am,, and the dependence of am on the lattice coordination number 2. For mixtures of nonassociating components, a quadratic dependence of 2 on temperature was chosen, as well as a linear dependence of am,, on temperature. Weidlich and Gmehling (1987) have modified the UNIFAC-FV model by introducing a quadratic temperature dependence for the group interaction parameters am,,, somewhat similar to one of the approaches described by Skjold-Jerrgensen et al. (1980). Leung and Badakhshan (1987) empirically modified the residual contribution to UNIFAC-FV by focusing also on the 1c, parameters. However, the temperature function in the denominator of the exponential in eq 11was expanded in linear fashion relative to a reference temperature, and the group interaction parameters am,,were considered to be temperature-independent. Leung and Badakhshan (1987) found that this empirical modification did not correct the erroneous temperature-dependent activity predictions for organic acid-water-toluene ternary mixtures. More recent modifications of the UNIFAC group contribution method by Larsen et al. (1987), Gupte and Danner (1987), and Park and C a r (1987) also address the concept of temperature-dependent interaction parameters and critically evaluate the potential for enhanced predictive capabilities resulting from the inclusion of such temperature dependences. In this study, temperature dependence has been incorporated also in the component specific volumes for the

288 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988

free-volume contribution (see eq 6 and 7) via thermal expansion coefficients. Patterson (1969, 1982) has emphasized that the strong unfavorable temperature dependence of the free-volume contribution to the Gibbs free energy of mixing can produce a lower critical solution temperature phenomenon (i.e., phase separation at elevated temperatures). Incidentally, the equilibrium phase stability of polystyrene and poly(viny1 methyl ether), below the LCST phenomenon, is attributed to a favorable free-volume contribution to the free energy of mixing, rather than strong intermolecular association between functional groups on dissimilar chains (Fredenslund, 1986). In this respect, successful predictions of temperature-dependent stability/instability in polymer-polymer blends via UNIFAC-FV should focus on the free-volume contribution to the activity of polymeric component i and seek methods to incorporate a strong positive/unfavorable temperature dependence in this term. From a quantitative viewpoint, the UNIFAC-FV numerical predictions of the specific Gibbs free energy of mixing for PS-PVME blends (Figure 1)are always negative (in sign) in the extreme limits of undiluted PS and undiluted PVME. These trends are consistent with theoretical equilibrium thermodynamic arguments which dictate that the slopes at the concentration limits should approach +m (for pure PS) and --OD (for pure PVME) (Patwardhan and Belfiore, 1986). Although our calculations in Figure 1have the proper qualitative shape at both extremes on the concentration axis, the numerical slopes are somewhat smaller, in absolute magnitude, than those dictated by theory. This apparent discrepancy is attributed to our inability to approach the extremes numerically on the Concentration axis in the UNIFAC-FV computational scheme. In this respect, we have interrupted all curves of Gibbs free energy of mixing versus concentration (at the concentration limits) presented in this paper. It is well-known that the compatibility of polystyrene and poly(viny1methyl ether) is influenced strongly by the choice of solvent used for film casting (Bank et al., 1971). Toluene, benzene, and tetrachloroethylene produce compatible mixtures, whereas chloroform, dichloromethane, and trichloroethylene produce phase-separated systems, both in solution and in solvent-free solid-state films (Su and Patterson, 1977). Our equilibrium thermodynamic calculations of the free energy of mixing for polystyrene and poly(viny1 methyl ether) are qualitatively consistent with the blend stability observed in films cast from benzene, toluene, or tetrachloroethylene. However, the UNIFAC-FV model is insensitive to the kinetics of solvent devolatilization and, hence, cannot predict the kinetic phase behavior of PS-PVME films cast from chloroform, dichloromethane, or trichloroethylene. UNIFAC-FV predictions in Figure 2 illustrate component molecular weight effects on the concentration-dependent Gibbs free energy of mixing for polystyrene and poly(viny1 methyl ether) at 373 K. Both curves in Figure 2 are characteristic of a completely miscible blend for which the diffusional stability criteria are satisfied over the entire concentration range of the binary mixture. Once again, our results are consistent with the experimental findings of Nishi and Kwei (1975), who have reported concentration-dependent and molecular weight dependent cloud points that are well above 100 "C for the component molecular weights chosen to calculate the Gibbs free energy of mixing via UNIFAC-FV. However, in addition to the extremely weak temperature dependence of the groupcontribution predictions, as discussed above, a second shortcoming of the UNIFAC-FV method is evident from

n

a

1:" z0x 1

b

>

c1

U

W

Z

w W W

U

L

v1 D

m0

0.2

0.0

0.4

0.6

0.8

1.o

WEIGHT FRACTION OF POLYSRRENE

Figure 2. Molecular weight effecta on the concentration dependence of the Gibh free energy of mixing at 373 K for.blends of polystyrene and poly(viny1 methyl ether). Calculations were performed on monodisperse molecular weight polymers: (a) PS (40 000) and PVME (10000); (b) PS (51000) and PVME (51500).

the relative magnitudes of the curves in Figure 2. The Gibbs free energy of mixing (at constant temperature and pressure) is more negative for the higher molecular weight blends and PS and PVME (see curve b). This result contradicts the idea of entropic stabilization, which is expected to provide a stronger contribution (negative in sign) to AG- when the component molecular weights are lower. Patterson (1982) illustrates this important feature (the decrease in ASmk as the molecular size of the components increases) for free energy of mixing curves that are characteristic of a generic binary mixture. The thermodynamic group contribution calculations discussed herein in terms of the specific Gibbs free energy of mixing have been extended to predict the polymerpolymer interaction parameter for the above-mentioned blends of polystyrene and poly(viny1methyl ether). Based on a modification of the Flory-Huggins lattice theory appropriate to binary mixtures of chain molecules, the dimensionless interaction Gibbs free energy of mixing, x'p~pvME (per repeat unit of polystyrene), was calculated from the activity of polystyrene in the blends (Olabisi et al., 1979; Hadziioannou et al., 1983), In ups = 1

and from the molar Gibbs free energy of mixing (Olabisi et al., 1979; Patterson, 1982; Solc et al., 1984; Shiomi et al., 1986),

1

~ P S ~ P V M E ~ P S X ' P S - P V M E(19)

In eq 18 and 19, 4i is the segment (or volume) fraction and

Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 289 0.00

I

iK -0.05W

L

:

-0.10

-

z -0.15

-

IL

Q

a

0k

i:L -0.20 -

0.0

0.2

0.4

0.6

0.8

1 .0

WEIGHT FWCTION OF POLYSTYRENE

Figure 3. Concentration dependence of the Flory-Huggins thermodynamic interaction parameter (per repeat unit of polystyrene) in blends of polystyrene and poly(viny1 methyl ether) a t 370 K. Calculations were performed on monodisperse molecular weight polymers: PS (40 000) and PVME (10000). (a) xPS-PVME based on the activity of polystyrene in the blends (eq 18); (b) xps-pvmbased on the Gibbs free energy of mixing (eq 19).

ri is the number of segments per chain molecule (e.g., degree of polymerization) of polymeric component i. In both cases, predictions of the dimensionless thermodynamic interaction parameter agree quite well with some of the experimental values of x ‘ ~ reported ~ - in~the ~ literature. Based upon the activity of polystyrene (eq 18), X’PS-~VME was predicted in the range -1.3 x to -4.1 X 10-1 (see the upper curve in Figure 3). In terms of the Gibbs free energy of mixing (eq 19) which does not restrict the interaction parameter to be concentration independent, X’PS-~VME was predicted in the range -3.2 x 10-1 to -4.2 X lo-’ (see the lower curve in Figure 3). As illustrated in Figure 3, the concentration dependence of the interaction parameter has been demonstrated at 370 K for a polystyrene monodisperse molecular weight of 40 000 and a poly(viny1 methyl ether) monodisperse molecular weight of 1OOOO. Koningsveld and Kleintjens (1971) and Zhikuan (1984) discuss the concentration dependence of the interaction parameter as a truncated power series based on the segment or volume fraction. Our predictions of the concentration-dependent polymer-polymer interaction parameter are self-consistent with the lattice model for AGmix,moler,given by eq 19, in which ?-pSX’p&.pVME (written here on a per molecule basis for monodisperse polystyrene chains) potentially exhibits dependence on temperature, concentration, and molecular weights of the components in the mixtures. However, the modified Flory-Huggins expression for the activity of polystyrene in the blends, given by eq 18, was derived from eq 19 (for AGmix,molar) under the restriction of a concentration-independent interaction parameter (Olabisi et al., 1979). Nevertheless, our predictions of x’pspvhlE(per repeat unit of polystyrene) are less than zero in all cases and compare well with the predominantly small negative values determined experimentally: -0.78 Ix’ I-0.17 reported by Kwei et al. (1974) from vapor sorption of benzene, -0.13 Ix’ I+1.47 re-

-2.41 00

1

02

,

~

04

,

, , 06

, 08

1 .o

WEIGHT FRACTION OF POLYSMRENE

Figure 4. Concentration dependence of the Gibbs free energy of mixing at 473 K for blends of polystyrene and poly(methy1acrylate). Calculations were performed on monodisperse molecular weight polymers: PS (100000) and PMA (80000).

ported by Su and Patterson (1977) from gas-liquid chromatography in the presence of various different vaporphase probe molecules, x’ = -4.1 X lo4 and -3.2 X reported by Hadziioannou and Stein (1984) from smallangle neutron scattering data in the solid state, -2.0 X I x’ I +2.0 X reported by Herkt-Maetzky and (1983) from small-angle neutron scattering ~Schelten ~ 5 x’ measurements on molten mixtures, and -6.0 X I+2.7 X reported by Shiomi et al. (1985) from osmotic pressure measurements in the presence of either toluene or ethylbenzene. To the best of our knowledge, a complete listing of all published thermodynamic interaction parameters for blends of polystyrene and poly(viny1 methyl ether) is given in Table 11. One should exercise caution in making comparisons among these tabulated values because of the dependence of xIPS-pVME on temperature, concentration, component molecular weights, polydispersities, and, most likely, experimental method which, in some cases, includes a third low molecular weight component. The UNIFAC-FV model calculations of x’pspvME (per repeat unit of polystyrene) presented herein were performed on hypothetical monodisperse molecular weight polymers (MWps = 40000; MWpVME = 10000) in the absence of a third low molecular weight component at 370 K. Polystyrene-Poly(methy1 acrylate). The concentration dependence of the specific Gibbs free energy of mixing, predicted by the UNIFAC-FV model for blends of polystyrene and poly(methy1acrylate), is illustrated in Figure 4 at a temperature of 473 K. The hypothetical monodisperse molecular weights were chosen as 100000 for polystyrene and 80000 for poly(methy1 acrylate). In this manner, the experimental miscibility study via melt titration by Somani and Shaw (1981) provides a basis for comparison with the UNIFAC-FV model predictions. Once again, the concentration dependence of the activity of polystyrene was calculated by using eq 1-11, including the modification of the combinatorial activity proposed

290 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 Table 11. Tabulation of Published Thermodvnamic Interaction Parameters for Blends of Polvstvrene and Polv(vinv1 methvl ether) ( X’PS-PVME)

-0.78

(X ’ P S - P V M E ) ~ ~ ~

-0.17

413

+1.47

-4.1 x 10-4 -6.0 X lo-’

-3.2

X 10

+2.7

X

’

10.’

temp, ”C

wt fraction of

30. 50

0.35, 0.45, 0.55, 0.65

40

0.45, 0.625

25 25

0.75 0.50

PS

25

molec wt of PS and polydispersity [q]”= 0.94 dL/g

molec wt of PVME and polydispersity

M , = 51 500

in benzene at 25 “C M , = 600, M,/M, < 1.1

M , = 10000

M,, = 71000 M,/M, = 1.09

M , = 46500 M , / M , = 2.12

M , = 1.6 x 105, M , = 2.1 x 105

M , = 7.4

X

lo4

(monodisperse)

-2.0

X

-0.41

-0.42

10.’

+2.0 X 10.’

-0.013

-0.32

154-190

97

91

0.35, 0.59, 0.75 (actually vol fraction)

M , = 47000, M,/M, = 1.06

M , = 99000, M , / M , = 2.12

0.01-0.99

M = 40000 (monodisperse)

M = 10000

M = 40000 (monodisperse)

M = 10000

0.01-0.99

(monodisperse)

(monodisperse)

comments Kwei et al. (1974); vapor sorption of benzene, x per segment of P S Su and Patterson (1977); gas-liquid chromatography, several vapor-phase probe molecules, x normalized to the size of the probe Hadziioannou and Stein (1984): SANS using deuteriated PS, x per repeat unit of P S Shiomi et al. (1985): osmotic pressure measurements in toluene or ethylbenzene; actually x ’ ~ ~ normalized ~ , ~ . ~ ~ to the size of the toluene solvent molecule Herkt-Maetzky and Schelten (1983): SANS using deuteriated PS, x per repeat unit of PS this study: x per repeat unit of PS, based on the activity of PS from UNIFAC-FV (the concentration dependence of x is not accounted for correctly) this study; x per repeat unit of PS, based on the Gibbs free energy of mixing from UNIFAC-FV, the Gibbs-Duhem equation, and Euler’s integral theorem (the concentration dependence of x is accounted for correctly in the modified Flory-Huggins model)

[ q ] = intrinsic viscosity.

by Thomas and Eckert (1984), whereas the activity of poly(methy1 acrylate) was determined from an integral form of the Gibbs-Duhem equation (similar to eq 16 for blends of PS and PVME). The free-volume contribution to the activity of polystyrene was calculated by setting the free-volume constant C1to the value of 660. As mentioned above, this value of C1 was determined for a hypothetical monodisperse molecular weight of 100000 using the entry for polystyrene. in Table I (C1/M1= 6.6 X The group contribution predictions for polystyrene and poly(methy1 acrylate) in Figure 4 indicate that the Gibbs free energy of mixing (at constant temperature and pressure) is less than zero and exhibits the appropriate curvature for a binary system that is completely miscible. These characteristics of the data generated by the UNIFAC-FV model contradict the fact that almost complete immiscibility is observed experimentally for this blend system via melt titrations a t 423 K (Somani and Shaw, 1981). However, an example from the literature illustrates the fact that the proper choice of solvent (naphthalene) together with a rapid freeze-drying procedure can produce a kinetically homogeneous solid-state blend of polystyrene and poly(methy1 methacrylate) (Shultz and Young, 1980), closely akin to the blend of interest herein. The dimensionless interaction Gibbs free energy of mixing, x’pS-pMA (per repeat unit of polystyrene), was calculated a t 473 K for the same hypothetical monodisperse molecular weight fractions of polystyrene and poly(methy1 acrylate) used to generate AGmixin Figure 4. Curve a in Figure 5 represents concentration-dependent predictions of the polymer-polymer interaction parameter based on the Flory-Huggins lattice theory expression for the activity of polystyrene (eq 18). Self-consistent prebased dictions of the concentration dependence of x’pspMA,

on the lattice theory model for the molar Gibbs free energy of mixing, are represented by curve b in Figure 5. As expected, predictions of a negative polymer-polymer interaction parameter from AGmix(curve b) in Figure 5) are consistent with the sign and curvature of the incorrectly predicted AGmix illustrated in Figure 4 for this polymer blend. Calculations of x’pS-pMA based on the activity of polystyrene (curve a in Figure 5) are predominantly less than zero, except at high concentrations of polystyrene (greater than 87 w t % PS) where the interaction parameter is positive. A discussion of this latter concentration dependence of x’pS-pMA is precluded by the fact that the UNIFAC-FV model is inconsistent with the equilibrium thermodynamic phase behavior observed experimentally for blends of polystyrene and poly(methy1 acrylate). Furthermore, literature citings of x ’ ~ are~ relatively - ~ ~ scarce. However, Somani and Shaw (1981) have reported a Tompa interaction parameter of +28 for polystyrene (this is equivalent to ~ P s x ’ p s - p M A , or the dimensionless interaction free energy of mixing per molecule of polystyrene chains [see the last term in eq 191) at a temperature of 423 K, poly(methy1acrylate) volume fraction of 3.4 X (this concentration represents the limit of solubility on the binodal curve), polystyrene weight-average molecular weight of 520 000, and poly(methy1 acrylate) weight-average molecular weight of 200,000 with a polydispersity of 3.16. The data of Somani and Shaw (1981) correspond to an interaction parameter of +5.6 x calculated as x’pS-pMA (per repeat unit of PS) assuming monodisperse chains of polystyrene. For comparison, curve a in Figure 5 reveals a UNIFAC-FV prediction of x ’ ~ = +6.2 ~ -X ~ at 473 K, poly(methy1 acrylate) weight fraction of 0.01, polystyrene monodisperse molecular weight of 100OOO, and poly(methy1 acrylate) monodisperse molecular weight of

~

~ ~

Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 291 7.0 I

0.10

5

0.05 h

t

>” (1

W

0.00 K d a

(1

Z -

z -0.05

xI b

a /’

I!

+

2C -0.10

>

li

(1

1

ffl

K w Z w

(1

W W

? -0.15

z -0.20 0

K

3

1 ‘-0.25 i:

w

0

c1

m -m

-0.30

-0.35

0.0

0.2

0.4

0.6

0.8

1.o

WEIGHT FRACTIOH OF POLYSTYRENE

F i g u r e 5. Concentration dependence of the Flory-Huggins thermodynamic interaction parameter (per repeat unit of polystyrene) in blends of polystyrene and poly(methy1 acrylate) a t 473 K. Calculations were performed on monodisperse molecular weight polymers: P S (100000) and PMA (80000). (a) xPs-PMA based on the activity of polystyrene in the blends (eq 18); (b) xPS-pMA based on the Gibbs free energy of mixing (eq 19).

80 000, with xIPS-pMA determined from the activity of polystyrene via eq 18. Once again, one should exercise caution when comparing polymer-polymer interaction parameters at different temperatures, concentrations, component molecular weights, and polydispersities. Poly(ethy1ene glycol)-Poly(acry1ic acid). As a final example, the concentration dependence of the specific Gibbs free energy of mixing, predicted by the UNIFAC-FV model for blends of poly(ethy1ene glycol) and poly(acry1ic acid), is illustrated in Figure 6 at a temperature of 350 K. The introduction of a potentially crystallizable component, poly(ethy1ene glycol), provides an additional complexity in the morphology of these blends that cannnot be described satisfactorily via group contribution methods. For this reason, the data in Figure 6 were generated at a temperature which is above the melting point of poly(ethy1ene glycol), where both components of the blend are present in the isotropic molten state. The hypothetical monodisperse molecular weights were chosen to be 3400 for poly(ethy1ene glycol) and 5000 for poly(acry1ic acid). Such relatively low molecular weight components enhance the potential for miscibility (in practice) via the nonnegligible contribution of entropic stabilization to AGmk, in addition to intermolecular association between these dissimilar blend components in the form of hydrogen bonds (Smith et al., 1959). Furthermore, these component molecular weights are commercially available from Scientific Polymer Products. This allows one to probe the miscibility of such blends from an experimental viewpoint and generate data against which the thermodynamic group contribution predictions can be tested. In this case, UNIFAC-FV was employed to calculate the concentration-dependentactivity of poly(ethy1ene glycol) in conjunction with the Staverman formula for the combinatorial contribution (eq 2). The free-volume contribution to the activity of poly(ethy1ene glycol) was calculated by setting the free-volume constant

0.0

0.2

0.4

0.6

0.8

1.o

WEIGHT FRACTION OF PEG

Figure 6. Concentration dependence of the Gibbs free energy of mixing a t 350 K for relatively low molecular weight blends of poly(ethylene glycol) and poly(acry1ic acid). Calculations were performed on hypothetical monodisperse molecular weight polymers: PEG (3400) and PAA (5000).

C1to the value of 32. This value of C1 was determined for a hypothetical monodisperse molecular weight of 3400 using the entry in Table I (C1/M1= 9.4 X lo9) for poly(ethylene oxide). The activity of the second component, poly(acry1ic acid), was obtained by invoking the GibbsDuhem equation in integral form. UNIFAC-FV predictions in Figure 6 for this strongly associating polymer pair indicate that AGmi, (at constant temperature and pressure) is greater than zero and exhibits curvature which is characteristic of an immiscible binary mixture over the entire range of physically measurable polymer concentrations. Such predictions contradict the mechanical results of Smith et al. (1959) which suggest the formation of an amorphous complex due to hydrogen bonding between the carboxylic acid groups of poly(acry1ic acid) and the ether oxygens of poly(ethy1ene glycol). In this respect, Skjold-Jargensen et al. (1980) mention that UNIFAC-FV is not capable of representing the equilibrium thermodynamic phase behavior of strongly associating or solvating systems. Calorimetric data (via DSC thermograms) are presented in Figure 7 for blends containing polydisperse samples of poly(ethy1ene glycol) (A?,, = 3400) and poly(acry1ic acid) (A?,, = 5000). The thermograms reveal a melting endotherm for undiluted poly(ethy1eneglycol) (PEG) and for the blend containing 7 2 wt % of the potentially crystallizable component (PEG). Furthermore, the DSC endothermic peak temperatures for thermograms c and d in Figure 7 are approximately 58 OC,indicating the failure of poly(acry1ic acid) to induce melting point depression in the blend containing a high concentration of PEG. In contrast, no PEG melting endotherm is observed in thermogram b of Figure 7 , which corresponds to the blend containing a relatively low concentration (27 wt % PEG) of the potentially crystallizable component. The DSC results presented in Figure 7 support the concept of a binary mixture that exhibits concentration-dependent

292 Ind. Eng. Chem. Res., Vol. 27, No. 2 , 1988

d 12

5I

51

-B

9P

IB

'.mp.-oi"re

!88 ,,a -0

128

138

,a

158

Figure 7. DSC thermograms for relatively low molecular weight blends of poly(ethy1ene glycol) and poly(acry1ic acid). Average molecular weights of the components are as follows: PEG (3400) and PAA (5000). The heating rate was maintained a t 5 OC/min on a Du Pont 1090 thermal analyzer. (a) Poly(acrylic acid), ( b ) 27 wt % poly(ethy1ene glycol), (c) 72 wt 70 poly(ethy1ene glycol), (d) poly(ethylene glycol). Table 111. Combinatorial, Free-Volume, and Residual Activities, Calculated by Using Equations 2-11 in the Text and the UNIFAC-FV Parameters, for Blends of Polystyrene, Component 1, and Poly(methy1 acrylate), Component 2 PS w t fraction In ulc In alFV In ulR 2.4 -217 0.1 -2.6 1.9 -151 0.2 -1.8 0.3 -1.4 1.4 -100 0.4 -1.1 1.0 -6 2 0.7 -35 0.5 -0.8 0.5 -17 0.6 -0.6 0.i -0.4 0.3 -6.5 0.8 -0.3 0.1 -1.4 0.9 -0.1 0.04 0.04

miscibility due to the crystallization tendency of one of the components (PEG). At high concentrations of PEG, blend morphologies include a PEG-rich crystalline phase containing vanishingly small amounts of the acidic component (as evidenced by the absence of PEG melting point depression) and a well-mixed amorphous phase (based on the mechanical results of Smith et al. (1959); and the solidstate carbon-13 NMR studies of Belfiore (1987)). At low concentrations of PEG, single-phase stability is dictated by the propensity for intermolecular hydrogen bonding (Smith et al., 1959) coupled with the absence of PEG crystallization (see thermogram b in Figure 7 ) . Discussion of the UNIFAC-FV Predictions. In this section, we focus attention on the numerical values associated with each contribution to the activity of polymeric component i in a nonassociating (non-hydrogen bonding) polymer-polymer blend. For illustrative purposes, we have chosen the polystyrene-poly(methy1 acrylate) system in which the UNIFAC-FV predictions discussed above are very discouraging, in comparison with experimental results. A close look at the concentration dependence of the combinatorial, free-volume, and residual activities of poly(styrene) in blends with poly(methy1 acrylate) (see Table 111) indicates that the natural logarithm of the residual activity is almost always large and negative and increases monotonically as the weight fraction of polystyrene increases. Consequently, the total activity of polystyrene increases monotonically as its weight fraction approaches unity, thus implying that the mixture is completely miscible. Based upon the criterion for chemical (diffusional)stability in a binary mixture, phase separation will persist, possibly over a restricted concentration range,

when the concentration dependence of the activity of one of the components in the mixture is negative, at constant temperature and pressure (Model1and Reid, 1983). Since the combinatorial activity (an entropic contribution) favors miscibility, the sum of the free-volume and residual contributions must outweigh the combinatorial term to correctly predict immiscibility in blends of polystyrene and poly(methy1 acrylate). Such a result will be obtained if In alRis greater than unity or, in some cases, if it is negative but is much smaller in absolute magnitude than those listed in the Table I11 results. The UNIFAC group interaction parametes umn have been determined from data for small molecules only. In a mixture of small molecules, dissimilar molecules and, therefore, all functional groups exist in close proximity to each other. This is not the case in polymer-polymer blends where severe restrictions due to chain connectivity are placed on the allowed conformational states of the backbone bonds. Hence, functional groups from dissimilar chains are prevented, in some cases, from participating in intermolecular interactions due to the chainlike structure of the polymer molecules. As a result, the intensity of intermolecular interactions between functional groups from dissimilar molecules is different in mixtures of small molecules than in polymer-polymer blends. The group interaction parameters am,,determined from a data base containing only small molecules are, therefore, not suited for estimating interactions between functional groups attached to dissimilar chain molecules in a blend. In addition, the somewhat coiled structure of a polymer molecule, particularly for incompatible mixtures, shields functional groups in its interior. For these reasons, it is suggested that the residual part of the activity is exaggerated in magnitude for polymer-polymer blends, and a separate set of group interaction parameters would be required to correctly predict the equilibrium phase behavior in mixtures of chain molecules. Finally, it is suggested that the inadequacy of eq 8 (using group interaction parameters appropriate to small molecules) for predicting the residual contribution to the activity of a chain molecule stems from the coefficients vk(,) which represent the number of functional groups of type k in polymeric component i. For example, these coefficients for the different functional groups which comprise the polystyrene repeat unit approach a value of 1000 (5000 for the aromatic CH group) for PS molecular weights on the order of lo5 (this hypothetical molecular weight of polystyrene was chosen for mixture calculations with poly(methy1 acrylate)). It is evident from Table I11 and the discussion of the UNIFAC-FV model that the combinatorial and free-volume contributions to the activity of polystyrene do not contain the inadequacies inherent in the residual activity. In particular, the factor Mi1 in eq 4 for the pure component volume (rJ and surface area (qJ parameters restricts these quantities from becoming exceedingly large for high molecular weight molecules. This is a consequence of the fact that r, and q, are defined on a per unit mass basis. Furthermore, the definitions of the surface and segment fractions, 0, and 4,, in eq 3 employ ratios of the q1and r,, respectively, thereby providing another internal restriction before substitution of the parameters (q,, O,, and &) into the combinatorial expression for the activity of polymeric component i. In summary, the UNIFAC-FV model fails to provide satisfactory predictions of the equilibrium thermodynamic phase behavior of polymer-polymer blends because of inadequacies in the residual activity expression when group interaction parameters appropriate to small molecules are

Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 293 employed. At present, no restriction is placed on the numerical value of the coefficients vk(i)in eq 8 which account for the number of functional groups of type k in component i. If the UNIFAC correlation, based on semiempirical models for small-molecule liquid mixtures, is indeed applicable to chain-molecule mixtures (most likely in a nonequilibrium state), then a method must be found to prevent very large values of uk(i)from giving unreasonable results for the residual activity.

Conclusions An attempt was made to predict the equilibrium thermodynamic phase behavior of polymer-polymer mixtures by modifying the combinatorial and free-volume terms of the UNIFAC-FV model. Staverman’s correction term was neglected in the expression for the Combinatorial activity, and Thomas and Eckert’s (1984) suggestion was implemented to eliminate the physically unrealizable positive combinatorial contribution. The free-volume constant for polymers was determined from the characteristic parameters of Flory’s equation-of-state model. However, the inapplicability of the group interaction parameters umn (determined from mixtures of small molecules) for polymer-polymer blends leads to an overly large residual activity. This prevents one from making accurate qualitative and quantitative predictions of the phase stabilityjinstability of amorphous polymer-polymer mixtures in their equilibrium state. Polymer blend morphologies discussed herein include miscible, immiscible, and partially miscible systems for which experimental phase behavior results from the literature (as well as data generated in this contribution) provide a basis for comparison with the UNIFAC-FV predictions. Calculations of the Flory-Huggins thermodynamic interaction parameter for blends of polystyrene and poly(viny1 methyl ether) a t temperatures below the lower critical solution phenomenon agree quite well with previously reported experimental thermodynamic results. However, UNIFAC-FV fails to predict (i) the high-temperature phase separation phenomenon in blends of polystyrene and poly(viny1 methyl ether), (2) the complete immiscibility (for all practical purposes) in blends of polystyrene with poly(methy1 acrylate), and (3) the intermolecular association and concentration-dependent miscibility in relatively low molecular weight blends of poly(ethy1ene glycol) and poly(acry1ic acid). In light of these shortcomings, it is suggested that the UNIFAC-FV group interaction parameters am,,for polymer-polymer blends be determined from liquid-liquid equilibrium data on chain-molecule, rather than small-molecule, mixtures.

Acknowledgment The authors are grateful to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for full support of this research through Grant 16208-G7. Seong Young Lee deserves recognition for preparation of the figures in this manuscript. It is always a pleasure to acknowledge the secretarial assistance of Laurie Lockhart and Sharon Patterson.

Nomenclature al = thermodynamic activity of component 1 u2 = thermodynamic activity of component 2

am = UNIFAC energetic interaction parameter between group m and group n Awk = van der Waals surface area of group k

b = adjustable parameter for the free-volume contribution to UNIFAC in eq 7 C1 = free-volume constant related to the number of external degrees of freedom per molecule of component 1 AG,,, = change in the Gibbs free energy due to mixing H, = area fraction of group m in the residual contribution of UNIFAC

AH,,, = enthalpy change due to mixing M , = molecular weight of component i MW, = molecular weight of component i; see eq 16 and 17 N , = number of moles of component i p = thermodynamic pressure P* = characteristic pressure (or potential energy density) in

Flory’s equation-of-state model q, = van der Waals surface area per unit mass of pure com-

ponent i

Q k = surface area parameter per mole of group k r, = van der Waals volume per unit mass of pure component i rps = number of chain segments per molecule of polystyrene

(or degree of polymerization) r p w = number of chain segments per molecule of poly(viny1 methyl ether) or degree of polymerization R = gas constant, 1.987 cal/(mol K) Rk = volume parameter per mole of group k AS,,, = entropy change due to mixing T = absolute temperature

T * = characteristic temperature in Flory’s equation-of-state model y, = specific volume of component i VI = volume parameter for component 1in the free-volume contribution to UNIFAC; see eq 7 = volume parameter for the mixture in the free-volume contribution to UNIFAC; see eq 7 Vwk = van der Waals volume of group k V* = characteristic volume (or the hard-core specific volume) in Flory’s equation-of-state model w, = weight fraction component i in mixture 2, = mole fraction of component i in mixture Z = lattice coordination number for the combinatorial contribution in UNIFAC

v,

Greek Symbols = number of functional groups of type k in a molecule of species i d5= segment (or volume) fraction of component i in mixture 8, = surface area fraction of component i in mixture r k = residual activity of group It in mixture rkU)= residual activity of group k in a reference solution of pure component 1 $mn = Boltzmann-type parameter in the residual contribution to UNIFAC which accounts for the interaction energy between groups m and n in the mixture x’pSPVME = polymer-polymer interaction parameter, per repeat unit of polystyrene, in the Flow-Huggins lattice theory for blends of polystyrene and poly(viny1methyl ether) Subscripts 1 = component 1 in mixture 2 = component 2 in mixture i = species i in mixture k = group k m = group m n = group n PS = polystyrene PVME = poly(viny1 methyl ether) PMA = poly(methy1 acrylate) Superscripts 1 = molecule of component 1 i = molecule of species i C = combinatorial contribution F V = free-volume contribution R = residual contribution Registry No. PS, 9003-53-6; PVME, 9003-09-2; PMA, 9003uk(l)

21-8; PEG, 25322-68-3; PAA, 9003-01-4.

I n d . Eng. Chem. Res. 1988,27, 294-303

294

Literature Cited Abrams, D. S.; Prausnitz, J. M. AZChE J. 1975,21,116. Bank, M.; Leffingwell, J.; Thies; C. Macromolecules 1971,4,43. Bates, F. S. Macromolecules 1985,18, 525. Belfiore, L. A. Polym. Prepr. (Am. Chem. SOC.,Diu. Polym. Chem.) 1987,28(2), 114. Belfiore, L. A,; Cooper, S. L. J . Polym. Sci., Polym. Phys. Ed. 1983, 21,2135. Beret, S.; Prausnitz, J. M. AIChE J . 1975,21,1123. Bondi, A. Physical Properties of Molecular Crystals, Liquids, and Glasses; Wiley: New York, 1968. Bonner, D. C.; Prausnitz, J. M. AZChE J . 1973,19,943. Brandrup, J.; Immergut, E. H. Polymer Handbook, 2nd ed.; Wiley: New York, 1975. Flory, P. J. Discuss. Faraday SOC.1970,49,7. Fredenslund, A. "Phase Equilibrium of Polymer Solutions by Group-Contribution, Part 2: Liquid-Liquid Equilibria". Personal communication, 1986. Fredenslund, A.; Gmehling, J.; Michelsen, M. L.; Rasmussen, P.; Prausnitz, J. M. Ind. Eng. Chem. Process. Des. Dev. 1977,16,450. Gmehling, J.; Rasmussen, P.; Fredenslund, A. Ind. Eng. Chem. Process Des. Deu. 1982,21,118. Gupte, P. A.; Danner, R. P. Ind. Eng. Chem. Res. 1987,26, 2036. Hadziioannou, G.;Stein, R. S. Macromolecules 1984,17,567. Hadziioannou, G.; Stein, R. S.; Higgins, J. Polym. Prepr. (Am. Chem. Soc., Diu. Polym. Chem.) 1983,24(2),213. Herkt-Maetzky, C.; Schelten, J. Phys. Rev. Lett. 1983,51(10), 896. Kammer, H. W. Acta Polym. 1986,37, 1. Kikic, I.; Alessi, P.; Rasmussen, P.; Fredenslund, A. Can. J . Chem. Eng. 1980,58, 253. Koningsveld, R.; Kleintjens, L. A. Macromolecules 1971,4,637. Kwei, T. K.; Nishi, T.; Roberts, R. F. Macromolecules 1974,7,667. Larsen, B. L.; Rasmussen, P.; Fredenslund, A. Ind. Eng. Chem. Res. 1987,26,2274. Leung, R. W. K.; Badakhshan, A. Ind. Eng. Chem. Res. 1987,26, 1593. Macedo, E. A.; Weidlich, U.; Gmehling, J.; Rasmussen, P. Ind. Eng.

Chem. Process Des. Dev. 1983,22,676. Magnussen, T.; Rasmussen, P.; Fredenslund, A. Ind. Eng. Chem. Process Des. Dev. 1981,20,331. Modell, M.;Reid, R. C. Thermodynamics and Its Applications; Prentice Hall: Englewood Cliffs, NJ, 1983; Appendix C, p 243. Nishi, T.; Kwei, T. K. Polymer 1975,16, 285. Nishi, T.; Wang, T. T.; Kwei, T. K. Macromolecules 1975,8, 227. Oishi, T.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1978, 17,333. Robeson, L. M.; Shaw, M. T. Polymer-Polymer MisciOlabisi, 0.; bility; Academic: New York, 1979; p 65. Park, J. H.; Carr, P. W. Anal. Chem. 1987,59, 2596. Patterson, D. Macromolecules 1969,2,672. Patterson, D. Polym. Eng. Sci. 1982,22,64. Phvs. Ed. Patwardhan, A. A.: Belfiore. L. A. J . Polvm. Sci... Polvm. 1986,24,2473. Shiomi. T.: Karasz. F. E.: MacKnicht. " , W. J. Macromolecules 1986. 19,2274. Shiomi, T.; Kohno, K.; Yoneda, K.; Tomita, T.; Miya, M.; Imai, K. Macromolecules 1985,18, 414. Shultz, A. R.; Young, A. L. Macromolecules 1980,13,663. Skjold-Jerrgensen, S.; Rasmussen, P.; Fredenslund, A. Chem. Eng. Sci. 1980,35,2389. Smith, K. L.; Winslow, A. E.; Petersen, D. E. Ind. Eng. Chem. 1959, 51, 1361. Solc, K.; Kleintjens, L. A.; Koningsveld, R. Macromolecules 1984,17, 573. Somani, R. H.; Shaw, M. T. Macromolecules 1981,14,1549. Staverman, A. J. Recl. Trav. Chim. Pays-Bas 1950,69, 163. Su,C. S.; Patterson, D. Macromolecules 1977,10, 708. Thomas, E. R.; Eckert, C. A. Ind. Eng. Chem. Process Des. Dev. 1984,23,194. Weidlich, U.; Gmehling, J. Ind. Eng. Chem. Res. 1987,26, 1372. Zhikuan, C. Polym. Commun. 1984,25, 21. Received for review October 6, 1986 Revised manuscript received October 20, 1987 Accepted October 21, 1987

PROCESS ENGINEERING AND DESIGN Sequential and Nonsequential Process Data Coaptation Burugupalli V. R. K. Prasad and James L. Kuester* Department of Chemical Engineering, Arizona State University, Tempe, Arizona 85287

Two process coaptation approaches, sequential and nonsequential, for data adjustment and estimation were investigated for a microcomputer implementation for an indirect liquefaction process. The steady-state Kalman filter, a sequential estimator, was found to be suitable for on-line process flow data estimation. The technique also serves as a means to carry out integration of the process data. The integrated data, free of systematic errors, could be used by the nonsequential technique. A reduced balance approach, using optimization routines, was needed for the nonsequential data coaptation problem. Adjustment of the composition variables and calculation of the process model parameters that satisfy material balance constraints are the principle benefits of the optimization approach. Several gross measurement error detection schemes, reported in the literature, were also tested for their suitability to the two techniques. A thermochemical conversion process to convert various biomass materials to diesel-type fuels has been under development at Arizona State University since 1975 (Kuester, 1984). An indirect liquefaction approach is used, i.e., gasification to synthesis gas (pyrolysis) followed by liquefaction of the synthesis gas (Fischer-Tropsch). A supervisory control system for this biomass indirect li0888-5885/88/2627-0294$01.50/0

quefaction process pilot plant was developed by utilizing a multitasking microcomputer, the IBM 9OOO. The control system features data acquisition, reduction, data coaptation, model development, simulation, and off-line/on-line optimization. Process data coaptation involves adjustment of the erroneous measurements and estimation of the unmeasured and unknown process variables. This is an 0 1988 American Chemical Society